We define a function which extends Gaussian hypergeometric series to the p-adic setting. This new function allows results involving Gaussian hypergeometric series to be extended to a wider class of primes. We demonstrate this by providing various congruences between the function and truncated classical hypergeometric series. These congruences provide a framework for proving the supercongruence conjectures of Rodriguez-Villegas.
- Gaussian hypergeometric series
- generalized hypergeometric series
- hypergeometric functions over finite fields
- p-adic gamma function